Condensed phase kinetics
• The kinetic theory of gases
involves ballistic motion
• Kinetics in liquids and other
condensed phases involves
diffusion
• What are the major
characteristics?
– Many molecules interacting at
once
– Short mean free paths
– No “memory” of momenta
Brownian motion
• Extreme case of
condensed phase
dynamics
• No memory of
momentum
• “Bath” provides:
– Random displacements
– Frictional damping
• Bath action related to
molecular collisions
Random walks
• Markov chain: no history
dependence
• Consider a discrete time
random walk in 1D:
– Equal probability of moving
left or right at each time step
– Position probability
distribution is binomial
– At a large number of steps,
probability becomes Gaussian
– Mean square displacement
grows linearly with time
• These are general features
of unbiased random walks
• Can be generalized to
continuous time and space
N!
⎛1⎞
P ( n+ , N ) =
⎜ ⎟
n+ !( N − n+ ) ! ⎝ 2 ⎠
N
⎛ 2n+2 ⎞
2
exp ⎜ −
∼
⎟
πN
⎝ N ⎠
∼
m2 ∼ N
2
⎛
m+ N)
(
1
exp ⎜ −
⎜
2π N
2N
⎝
1D random walk
10000 trajectories
10000 trajectories
40
0.2
30
0.15
0.1
20
0.05
x
<x>
10
0
0
-0.05
-10
-0.1
-20
-30
-0.15
0
100
200
300
400
n
500
600
700
800
-0.2
10000 trajectories
800
10
0
x 10
100
200
300
5
400
n
500
600
700
800
10000 trajectories
9
700
8
600
7
<x 2>
500
6
400
5
4
300
3
200
2
100
0
1
0
100
200
300
400
n
500
600
700
800
0
-150
-100
-50
0
50
100
150
Diffusion coefficients
• What is the mean
squared
displacement per
unit time?
• For unbiased
random walks, it’s
pretty simple
• Diffusion
coefficients also
have microscopic
interpretation…
m→
x, y , z
t
,N →
l
τ
P1D ( x, t ) = ( 4π Dt )
x2
1D
2D
2D
−1
⎛ x2 + y 2 ⎞
exp ⎜ −
⎟
4 Dt ⎠
⎝
= 4 Dt
P3D ( x, y, z , t ) = ( 4π Dt )
x2 + y2 + z 2
⎛ x2 ⎞
exp ⎜ −
⎟
Dt
4
⎝
⎠
= 2 Dt
P2D ( x, y, t ) = ( 4π Dt )
x2 + y2
−1 2
= 6 Dt
−3 2
⎛ x2 + y 2 + z 2 ⎞
exp ⎜ −
⎟
4
Dt
⎝
⎠
Langevin equation
• Newton’s equation with
extra terms
• Random force
dv ( t )
m
= −ζ v ( t ) + f ( t )
dt
f (t ) = 0
– Energy added by bath
– Mean force is zero
• Viscous drag
– Energy dissipated by
bath
– Related to velocity decay
times
• The viscous drag is
related to the diffusion
coefficient
m
d v (t )
= −ζ v ( t )
dt
v ( t ) = v ( 0 ) e −ζ t m
Viscosity and diffusion
• Solve Langevin
equation for mean
squared position
– Multiply by x and
rearrange
– Use the fact that the
mean squared velocity is
kT (Maxwell dist.)
– Solve for <xv> with
particle initially at origin
– Relate to <x2> and solve
m
d
( xv ) = −ζ xv + m v 2
dt
= −ζ xv + k BT
xv =
k BT
ζ
(1 − e
−ζ t m
)
kT
1 d 2
x = B (1 − e −ζ t m )
ζ
2 dt
x2 =
2 k BT ⎛ m
−ζ t m ⎞
t
1
−
−
e
(
)
⎜
⎟
ζ ⎝ ζ
⎠
Viscosity and diffusion
• Two time scales for mean
squared displacement
– Related to “collisional time”
– At short times, the motion is
ballistic
– At long times, the motion is
Brownian
• Velocity doesn’t matter
• RMSD can be related to
diffusion coefficient
• The friction and diffusion
coefficient are related!
x2 =
τ=
x2 =
lim x 2 =
t τ
2 k BT ⎛ m
−ζ t m ⎞
1
−
−
t
e
(
)⎟
ζ ⎜⎝ ζ
⎠
m
ζ
2 k BT
ζ
t τ
12 2
2 k BT
ζ
= 2 Dt
D=
−t τ
k BT 2
t
m
= v2
lim x 2 =
(t − τ (1 − e ))
k BT
ζ
t
t
The Stokes-Einstein relationship
• Stokes law provides an
expression for the
viscous drag around
various simple shapes
• This can be combined
with the MSD derivation
above
• The final relationship is
called the StokesEinstein relation
ζ = 6πη a
D=
k BT
ζ
k BT
=
6πη a
Mass action kinetics
• The rate of change in concentration or probability
• Describes a variety of phenomena: chemical
reactions, binding events, etc.
• Relates changes in concentrations with time to
(powers of) species concentrations
• Assumes large numbers of species
• Ignores fluctuations due to small copy numbers
aA + bB
k1
k2
P
dcP ( t )
a
b
= k1c A ( t ) cB ( t ) − k2 cP ( t )
dt
Michaelis-Menten kinetics
• Three basic reactions
– Substrate-enzyme association
– Substrate-enzyme dissociation
– Catalysis and product-enzyme dissociation
• Steady-state assumption used below
E+A
kon
koff
EA ⎯⎯→ E + P
kcat
koff
koff + kcat
Kd =
, KM =
kon
kon
kcat cE ( 0 ) cS ( t )
vss =
K M + cS ( t )
What is a diffusion-limited reaction?
• Consider a reaction where the chemical step is
instantaneous
– All reactions which form ES complex go to products
– The rate-limiting aspect of the reaction is binding to the enzyme
• Assume low substrate concentrations
E+A
kcat
kon
koff
koff , cS ( t )
kcat
EA ⎯⎯→
E+P
kcat
kcat
, KM ≈
kon
kon
kcat cE ( 0 ) cS ( t )
≈ kon cE ( 0 ) cS ( t )
vss ≈
kcat
+ cS ( t )
kon
Diffusion-limited reactions
•
Typical diffusional encounter rate is 109
to 1010 M-1 s-1
– There are lots of caveats, exceptions,
etc.: protein flexibility, electrostatics,
limited reaction surface
•
Smoluchowski encounter rate:
– Assumes spherical symmetry
– Based on solution of PDE
– No interactions: proportional to sum of
diffusion coefficients and separation
– Interactions: related to integral of
potential
•
•
Thought to represent evolutionary
pressure
Examples
– Superoxide dismutase
– Acetylcholinesterase
– Barstar-barnase
w( r ) k B T
⎡ e
⎤
kD ( R ) = ⎢ ∫
dr ⎥
2
⎣ R 4π r D ( r ) ⎦
k D0 ( R ) = 4π DR
∞
−1
Methods for diffusional encounter
simulations
• Discrete methods
– Langevin dynamics
– Brownian dynamics
– Monte Carlo
• Continuum methods
– Fokker-Planck
– Smoluchowski equation
Discrete simulations of binding events
• Brownian dynamics
• Use as normal dynamics methods
– Integrate stochastic equations of motion
– Average: configurations, thermodynamics, etc. (nothing
that depends on viscosities!)
• Use as encounter simulation method
First-order BD integration
• Calculate
– Diffusion coefficient gradient
– Potential of mean force gradient
– Random displacement
• Works for large time steps provided the gradients don’t
change (much)
• Position components can be x, y, z – or separate particle
coordinates
• Coupling between particle diffusion components:
hydrodynamic interactions
ri ( t + Δt ) = ri ( t ) + Δt ∑
j
Ri ( Δt ) = 0
Ri ( Δt ) R j ( Δt ) = 2 Dij Δt
∂Dij ( t )
∂rj
∂W ( t )
− Δt ∑ Dij ( t )
+ Ri ( Δt )
∂rj
j
BD for encounter rate calculation
• Assumptions:
– Low enzyme and substrate concentrations (no enzymeenzyme or substrate-substrate interactions)
– Diffusion control
– Implicit solvent
• Basic idea: what is the probability that two
molecules started at distance b will encounter one
another rather than wandering off to infinity?
BD for encounter rate calculation
• BD trajectory:
– Start two molecules at a
separation b where the
potential is centrosymmetric
– Integrate BD equation of
motion until
• Molecules satisfy reaction
criteria
• Molecules exceed separation
distance q
• A maximum number of steps
are taken
• Perform multiple BD
trajectories:
– Accumulate collision
frequencies
– Statistics are noisy; multiple
runs needed!
Figure from: Northrup SH, Allison SA,
McCammon JA. J Chem Phys 80 (4)
1517-24, 1984.
BD for encounter rate calculation
BD for diffusion-limited reactions
• Collision frequencies can
be transformed into rates
• Think: flux through reactive
site!
• If all collisions result in
reaction (diffusion-limited),
rate is related to:
– Rate of diffusion to separation
b (can use Smoluchowski
formula)
– Collision frequency
– Probability that trajectories
leaving q returns to b
kD ( b ) β
k=
1 − (1 − β ) Ω
w( r ) k B T
⎡ e
⎤
kD ( b ) = ⎢ ∫
dr ⎥
2
⎣ b 4π D ( r ) r
⎦
∞
∞
Ω=
e
w( r ) k B T
∫ 4π D ( r ) r
q
∞
e
dr
2
dr
w( r ) k B T
∫ 4π D ( r ) r
b
2
−1
BD for diffusion-influenced reactions
• If only some collisions result in reaction
(probability α), rate is related to:
– All of above
– Reaction probability α
– Probability Δ that unsuccessful encounter results in later
collision
⎡
⎤
β
α kD ( b ) ⎢
⎥
1 − (1 − β ) Ω ⎦
⎣
k=
⎧⎪
⎡
⎤ ⎫⎪
β
1 − (1 − α ) ⎨Δ + (1 − Δ ) ⎢
⎥⎬
⎪⎩
⎣1 − (1 − β ) Ω ⎦ ⎪⎭
Interactions in BD calculations
•
Forces
– Long-range influences only
– Electrostatics: approximate
charge-field calculations
Filig ≈ qilig E prot
• Poisson-Boltzmann calculation for
protein, charge model for ligand
• No desolvation
• Little “internal dielectric” screening
(some effective charge methods)
•
Diffusion coefficients
– Should include rotation, translation,
and configuration changes
– No hydrodynamic interactions
• Probably OK for small ligands
• Stokes-Einstein isotropic diffusion
coefficients
• Coefficients do not depend on
distance or configuration
– Hydrodynamic interactions
• Include water-mediated effects
• Oseen and other (approximate)
analytic forms
• Configuration- and distancedependent
αβ
Dij
1 − δ ij
k BT ⎛ δ ij
≈
⎜ I+
cπη ⎜⎝ ai
2 Rij
⎧ai + a j
Rij = ⎨
⎩ rij
⎛ rij rij
⎜⎜ I + 2
rij
⎝
rij < ai + a j
rij ≥ ai + a j
⎞⎞
⎟
⎟⎟ ⎟
⎠⎠
Application to acetylcholinesterase
•
•
•
Hydrolytic enzyme in
neuromuscular junction
Subject of extensive
computational (BD) and
experimental study
Properties:
– Diffusion-limited catalysis
– Long, narrow active site
gorge
– Significant electrostatic
influences
AChE/TMA binding
• Binding of neurotransmitterlike molecule to
acetylcholinesterase
• Diffusion-controlled binding
• Significant dependence on
[NaCl]
• Sensitivity to charged
residues
Figures from: Tara S, et al. Biopolymers 46 (7)
465-74, 1998.
DHFR-TS substrate channeling
•
•
Bifunctional enzyme:
thymidylate synthase
produces dihydrofolate used
by dihydrofolate reductase
Electrostatic steering between
active sites enhances
efficiency
– Reduces diffusional
broadening
– Does not “direct” between
sites
Figures from: Elcock AH, et al. J Mol
Biol 262, 370-4, 1996.
Protein-protein encounter
•
•
•
Same basic procedure as before
Reaction criteria are harder to evaluate – often a
variable in the simulation
Effective charge method
–
–
•
Use full electrostatic grid for one molecule
Use a smaller number of charges: termini and
charged residues
Usually neglect:
–
–
–
–
–
Desolvation terms
Hydrodynamic interactions (with exceptions)
Ion relaxation
Flexibility
Substrate-substrate interactions
Figures from: Gabdoulline RR,
Wade RC. J Phys Chem 100
3868-78, 1996 and ibid.
Methods 14 329-41, 1998.
Actin polymerization
• BD simulations of actin
polymerization
• Reproduced experimental
observation of faster
polymerization at “barbed”
filament end
• Implicated electrostatics in
faster binding to barbed end
• Also observed effect of ADFcofilin on polymerization
Figures from: Sept D, Elcock
AH, McCammon JA. J Mol
Biol 294 (5) 1181-9, 1999.
Lots of protein-protein association rates
•
Systems studied:
–
–
–
–
•
•
•
•
Barstar-barnase
AChE-Fas2
Cyt C peroxidase-Cyt C
HyHEL antibodies and lysozyme
Agreement with experiment is good
Antibody/lysozyme and AChE-Fas2 rates
overestimated: not diffusion-limited?
Electrostatics aren’t always helpful!
Figures from: Gabdoulline RR, Wade RC.
J Mol Biol 306 1139-55, 2001.
Continuum diffusion simulation methods
• Discrete methods
– Solve stochastic ODEs
– Provide atomic problem resolution
– Facilitate integration of stochastic
phenomena
– Software: MCell, UHBD, etc.
• Continuum methods
– Solve deterministic PDEs
– Bridge larger length scales
– Facilitate integration of continuum
mechanics phenomena
– Software: SMOL
Continuum diffusion motivation
• Demonstrate:
– Accurate description of enzyme binding kinetics (steadystate and time-dependent)
– Simulation of synapse electrophysiology
– Extreme adaptivity of methods to bridge length scales
• Long-term goals:
– Integrate continuum and discrete methods
– More complete description of cellular-scale processes
Smoluchowski equation
Concentration
change over time
Diffusion term
Drift term
∂ρ ( x)
= ∇ ⋅ J ( x) = ∇ ⋅ D( x)[∇ρ ( x) + βρ ( x)∇W ( x)]
∂t
Flux
Diffusion
External
coefficient
ρ (x) = ρ
“Bulk” boundary condition
ρ ( x) = 0
Reactive boundary condition
n( x ) ⋅ J ( x ) = 0
Reflective boundary condition
k (t ) = ∫ J ( s ) ⋅ n( s )ds
The observable: the time-dependent
rate constant
potential
Advances and outlook
• Receptor flexibility
• Detailed binding
mechanisms
• Imperfect reactivity;
calculate “re-entrant”
trajectories
HEL-antibody flexibility in lysozyme binding from
Gabdoulline RR, Wade RC. J Mol Biol 306 113955, 2001.
Camphor release pathway from
cytochrome P450 from Guallar lab.
Advances and outlook
• Cellular simulations
• Proteomics-scale
interactions
• Crowded
environments
Virtual Cell schematic from Slepchenko BM,
Schaff JC, Carson JH, Loew LM. Annu Rev
Biophys Biomol Struct 31 423-41, 2002.
Crowding and GroEL simulation from Elcock AH.
PNAS 100 (5) 2340-4, 2003.